MLA 31500

Natural Sciences Elective

 

Order and Chaos in the Natural World

 

Spring Quarter 2014

 

CLASS NOTES

FIFTH CLASS

April 29, 2014

 

 

I.      STEWART, CHAPTER 9: SENSITIVE CHAOS

 

1.     According to Stewart, what was the conclusion of Ruelle and Takens regarding the Landau or Hopf-Landau scenario for the onset of turbulence?

 

2.     What positive claim did Ruelle and Takens make regarding the onset of turbulence.

 

3.     One can think about several questions regarding the experiment on Couette-Taylor flow that was performed by Harry Swinney and Jerry Gollub.  What was the physical arrangement of the experiment?  What was the experimental procedure?  What were the results, and how did those results differ from expectations?

 

4.     In what respects did the experiments of Harry Swinney and Jerry Gollub support the claims of Ruelle and Takens.

 

5.     An important test of the picture of Ruelle and Takens is to exhibit a strange attractor in the results of a computer experiment on some model of a dynamical system.  Ruelle, Takens and Packard proposed a scheme for using data for this purpose.  Suppose that your data were the output of a computer solving the Lorenz equations.  How would you organize and plot the data in order to exhibit the Lorenz attractor.

 

6.     Likewise, how do you organize and collect the data from real measurements of a dripping faucet in order to show that there is a strange attractor in the dynamics.

 

7.     What seem to be the most important issues in this chapter for Stewart?

 

 


II. GLEICK, pp 121-153: STRANGE ATTRACTORS

 

1.       The work of David Ruelle and Floris Takens is portrayed in GleickÕs account as if it was an effort to invent the concept of a strange attractor.  In this connection, how do they represent the behavior of a dynamical system (e.g., a turbulent fluid)?  What aspects of the behavior of such a system and the occurrence of chaos were they seeking to represent in a strange attractor?  How does this relate to Stephen SmaleÕs horseshoe map?

 

2.       What are the attributes of the Lorenz attractor (see the figures on pages 28 and 140 and the plate opposite page 114) that make it a strange attractor?

 

3.       Consider the Henon map (pages 149-150).  How do we know that the representative point of the system in the phase space has reached an attractor?  And what is the attractor in that case?

 

 

 

LINKS:

 

Return to Course Page: mla315spring2013.html

 

Return to Peter Vandervoort's Home Page:  pov.html

 

Go to the home page of the Department of Astronomy and Astrophysics

of the University of Chicago:  http://astro.uchicago.edu/